Question

Determine whether the inverse of \(f\) is a function. Then find the inverse. $$f(x)=\frac{5}{x}-6$$

Short answer

The inverse of \(f\) is a function and it is \(f^{-1}(x) = \frac{5}{x + 6}\).

Step-by-step solution

Rewrite the Function

To start, rewrite \(f(x) = \frac{5}{x} - 6\) as \(y = \frac{5}{x} - 6\).

Swap \(x\) and \(y\)

The next step is to interchange \(x\) and \(y\), resulting in \(x = \frac{5}{y} - 6\).

Solve for \(y\)

Next, solve for \(y\) by first adding 6 on both sides to get \(x + 6 = \frac{5}{y}\) and then taking reciprocals to get \(y = \frac{5}{x + 6}\) which is the inverse function.

Check if the Inverse is a Function

The inverse function will pass the horizontal line test, there is only one \(y\) for every \(x\), hence the inverse is also a function.

Key concepts

Horizontal Line Test

The horizontal line test is a simple way to determine if a function's inverse is itself a function. It works by drawing horizontal lines through the graph of the original function. If any horizontal line crosses the graph at more than one point, the inverse won't be a function. This is because multiple points of intersection mean that a single output could map to multiple inputs in the inverse function.
Understanding how this test relates to the original function is key when assessing its inverse. For example, in the function \( f(x) = \frac{5}{x} - 6 \), no horizontal line intersects the curve at more than one point. This shows that its inverse is indeed a function.
In essence, the horizontal line test helps you quickly ascertain the functional status of an inverse, saving you from setting up and solving equations when unnecessary.

Function Notation

Function notation is a way to denote functions in an expressive and clear manner. In mathematics, functions are often represented as \( f(x) \), where \( f \) indicates the function and \( x \) the variable. This notation helps communicate the codependence of elements within the function, making it more manageable.

• Using function notation makes it simple to symbolize operations on the function, such as finding derivatives or integrals.
• It helps in substituting inputs to see the resulting outputs, such as in inverse functions, where \( f^{-1}(x) \) represents the inverse function of \( f(x) \).

For the original function \( f(x) = \frac{5}{x} - 6 \), transitioning to \( y = \frac{5}{x} - 6 \) is simply switching from the convenient function notation to a more equation-focused approach that facilitates finding the inverse.

Solving Equations

Solving equations involves finding the values of the variables that make the equation true. In the context of inverse functions, solving equations becomes crucial when finding the inverse. Here, after swapping \( x \) and \( y \) to change perspectives, you engage in algebraic manipulation to isolate \( y \).

• Start by applying operations like addition or subtraction to both sides, as seen in \( x + 6 = \frac{5}{y} \).
• Take reciprocals or other mathematical steps needed to further isolate the dependent variable, resulting in \( y = \frac{5}{x+6} \).

This step-by-step algebraic journey allows you to find the inverse function. It underscores the application of basic algebraic skills and logical reasoning to switch a function into its inverse.

Reciprocal Functions

Reciprocal functions involve expressions of the form \( \frac{1}{x} \), where the input and output have an inverse multiplicative relationship. The given function \( f(x) = \frac{5}{x} - 6 \) utilizes a reciprocal form within its structure, making its understanding vital to working out the inverse.
To refresh, a reciprocal essentially means flipping the roles of numerator and denominator. For instance, the reciprocal of \( x \) is \( \frac{1}{x} \). When solving for inverses, taking reciprocals often features prominently, especially when tackling rational functions like the given one.

• In finding the inverse, you swap \( x \) with \( y \) and bring these reciprocal relationships into play, as shown when isolating \( y \).
• Reciprocal functions highlight the intertwined nature of multiplication and division, pivotal in both understanding function behavior and manipulating equations.

These insights into reciprocal concepts enhance comprehension of how inverse functions are derived from original functions, particularly those of a rational nature.